# Conjugacy-separable implies residually finite

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., conjugacy-separable group) must also satisfy the second group property (i.e., residually finite group)

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## Contents

## Statement

Any conjugacy-separable group is a residually finite group.

## Definitions used

### Conjugacy-separable group

`Further information: Conjugacy-separable group`

A group is termed **conjugacy-separable** if given any elements that are not conjugate in , there exists a normal subgroup of finite index such that the images of in are not conjugate.

### Residually finite group

`Further information: Residually finite group`

A group is termed **residually finite** if given any non-identity element , there is a normal subgroup of finite index in such that is not in .

## Proof

**Given**: A conjugacy-separable group , a non-identity element .

**To prove**: There is a normal subgroup of finite index in such that is not in .

**Proof**: Let be the identity element. Since no non-identity element is conjugate to the identity element, and are in distinct conjugacy classes. Thus, there exists a normal subgroup of finite index such that the image of is not conjugate to the image of . In particular, this implies that is not contained in , so we are done.